Impact of Antisite Disorder on the Resistivity of Strontium Ferromolybdate Ceramics

Abstract

In this work, we consider the influence of antisite disorder, e.g., Fe ions on Mo sites, Fe_Mo, and vice versa, Mo_Fe, on the resistivity of strontium ferromolybdate ceramics fabricated by the solid-state reaction method. Strontium ferromolybdate ceramics fabricated via solid-state reactions exhibit a low-temperature minimum resistivity owing to the interplay between the bulk metallic resistivity of the grains, which increases with temperature and becomes dominant at higher temperatures, and an intergrain tunneling mechanism of charge carrier conduction, which leads to a decrease in conductivity with decreasing temperature in the low-temperature region. The parameters of the bulk metallic resistivity and fluctuation-induced intergrain tunneling were determined by fitting the experimental data to these resistivity models. The impact of antisite disorder on the resistivity parameters was considered. It turns out that antisite disorder affects the effective barrier height of intergrain tunneling and the effective values of the barrier width and the barrier area. Disorder increases the effective barrier height for intergrain tunneling, increases its barrier width, and decreases the effective barrier area of nanosized barriers. The results are discussed using experimental data available in the literature.

1. Introduction

Strontium ferromolybdate double perovskite (Sr₂FeMoO_6−δ, SFMO) is a promising candidate for magnetic electrode materials for room-temperature spintronic applications because it exhibits a half-metallic character, for example, a theoretically 100% spin polarization, a high Curie temperature (note that ferrimagnets should be operated in their ordered magnetic state below the Curie point), and remarkable low-field magnetoresistance (LFMR) [1]. Nevertheless, SFMO has not yet been widely applied in spintronics. This is attributed to the low reproducibility of its magnetic properties, which partially originates from its strong dependence on the ordering degree of Fe and Mo ions in both the B′ and B″ sublattices of the double perovskite A₂B′B″O₆.

In SFMO, the dominant point defects are Fe and Mo antisite defects, for example, Fe ions on Mo sites—Fe_Mo, and Mo ions on Fe sites—Mo_Fe. This leads to antisite disorder (ASD) [1], which occurs when a B′-site ion occupies the wrong sublattice B″, and vice versa. Consequently, ASD in stoichiometric SFMO is characterized by the formation of Fe_Mo and Mo_Fe defect pairs, creating Fe-Fe and Mo-Mo nearest neighbors (Figure 1).

In stoichiometric SFMO, the fraction of ions B′ (or B″) on the incorrect sublattice is related to the order parameter P [3]:

$$P=(1-2ASD).$$

(1)

Here, the value of ASD varies from 0 (corresponding to a complete order) to 0.5 (describing a completely random Fe-Mo site occupancy). Experimentally, the order parameter P is determined using the site occupation derived from the Rietveld analysis [4]:

$$P=2g_{Fe}-1,$$

(2)

where g_Fe denotes the occupancy of Fe ions at their correct sites.

Antisite disorder affects both the magnetic and electronic properties of a material. Whereas the Fe-Mo-Fe neighbors in the regular SFMO lattice couple ferrimagnetically, the Fe-Fe neighbors at the antisites are antiferromagnetic [5]. This lowers the expected saturation magnetization according to [6]

$$M_s=M_s(ASD=0)-b\cdot ASD=M_s(ASD=0)-b\cdot (0.5-P_M/2).$$

(3)

The value of the saturation magnetization M_s is given in Bohr magnetrons per formula unit, M_s(ASD = 0) = 4 µ_B/f.u. for a theoretical value of b = 8 while empirical values of b range between 6.56 and 10 [1], and P_M is the order parameter determined by the magnetic field dependences of magnetization.

Moderate levels of ASD could be the cause of sufficient increase in the LFMR effect. Here, the application of an external magnetic field leads to a suppression of spin disorder in the magnetically frustrated areas around the antisite defects [7]. This is also in agreement with the findings in [8], where an increase in the magnetoresistance (MR) was observed when ASD was increased from 0.02 to 0.09. As pointed out by Sarma et al. [9], a further increase in the Fe/Mo disorder level would lead to the loss of spin polarization. This, in turn, means that all the contributions to the MR associated with the half-metallic character vanish as the ASD levels increase, in agreement with the results in [7]. In this way, a narrow concentration of ASD (ASD ≈ 0.1) reveals where the LFMR shows a maximum.

For massive ASD levels, the ferromagnetic state of the ordered material can even transform into a spin-glass-like response [5,10].

The conjunction of locally ordered regions forms magnetic nanodomains [8,11]. Therefore, antisite disorder gives rise to neighboring Fe-O-Fe regions and, thus, to the formation of antiphase boundaries (APBs) [8,12]. APBs are growth defects that occur at the interface between two coherently oriented crystallites, where the stacking sequence of FeO₆ and MoO₆ octahedra initiates from opposite cation sites. In general, APBs are characterized by a crystallographic shear vector R that describes the relative displacement of the two parts of the crystal on either side of the interface. In our case, the value of this lattice shift is equal to half a lattice parameter. In SFMO, the APBs coincide perfectly with the nanodomain boundaries [11]. APBs locally interrupt the ferromagnetic coupling between Fe and Mo ions and the indirect Fe-O-Mo-O-Fe exchange interaction occurring in defect-free structures, leading to new magnetic exchange interactions that are not present in defect-free materials [1]. The presence of APBs in the YBa₂Cu₃O₇ sample favors oxygen in-diffusion and, therefore, ionic conductivity [13].

ASD usually increases the resistivity of SFMO [1,14,15,16]. This originates from (i) the strong dependence of the mixed Fe²⁺/Fe³⁺ fluctuating mixed valence state on cationic disorder [17,18], (ii) the larger extent of magnetic scattering processes at antisites disturbing the periodicity of the lattice potential [15], and (iii) the possibility of trapping and recombination of charge carriers by like-neighbor bonds at APBs [19].

In polycrystalline highly ordered SFMO ceramics (ASD = 0.10…0.13) prepared at high temperatures, which comprise a minor impurity phase of Fe, the increase in resistivity was attributed to decreasing grain size, leading to more grain boundaries [20]. Samples of SFMO thin films deposited by pulsed laser deposition, consisting of spurious insulating SrMoO₄ [15], showed an increase in the residual resistivity.

SFMO thin films with varying ASD deposited by pulsed laser deposition using different background gases [14] exhibit semiconducting behavior, the temperature dependence of which was fitted to the Mott variable-range hopping model [21]:

$$\rho ={\rho }_0\exp {\left({\displaystyle \frac{T_M}{T}}\right)}^{1/4},$$

(4)

where ρ₀ is the pre-exponential factor and T_M is the characteristic Mott temperature. The same resistivity behavior was observed in SFMO ceramics at temperatures near room temperature (280–300 K) [22]. Here, a larger T_M value implies a higher disorder and/or large tilting of the B′O₆ or B″O₆ octahedra. Note that the parameter T_M can also be considered an effective energy barrier between the localized states [23]. Mott variable-range hopping was also obtained in polycrystalline Sr₂FeMo_1−xV_xO₆ (x = 0–0.5) ceramics [18]. In this case, the temperature dependence of the resistivity of the undoped SFMO (x = 0) sample shows metallic behavior. The origin of the metallic resistivity of SFMO is the above-mentioned mixed valence Fe²⁺/Fe³⁺ state, resulting in a +2.5 fluctuation mixed valence [17]. This mixed valence state is influenced by any substitution of the cations. The substitution of Mo by V introduces paramagnetic ions (except V⁵⁺) [24], making variable-range hopping plausible.

Similar to the SFMO ceramics synthesized by the solid-state reaction method in our previous study [25], the samples in this work exhibited a temperature of minimum resistivity. This minimum is caused by the interplay of an intergrain tunneling mechanism of charge carrier conduction, which leads to an increase in resistivity with decreasing temperature in the low-temperature region, and a bulk metallic resistivity of the grains, which increases with temperature and becomes dominant at higher temperatures. The intergrain tunneling is attributed to fluctuation-induced tunneling (FIT) [26] with a resistivity given by

$$\rho (T)={\rho }_{FIT}\exp \left({\displaystyle \frac{T_1}{T_0+T}}\right).$$

(5)

where the temperatures T₁ and T₀ and the resistivity ρ_FIT are model fitting parameters. This model was developed to analyze the resistivity mechanism in a system composed of metallic particles separated by thin dielectric layers (a few nanometers thick). In particular, it aims to describe the mechanisms of electrical transport between carbon particles embedded within a polyvinyl chloride matrix. Polycrystalline SFMO ceramics can also be considered as a system in which conducting SFMO grains are spatially separated by dielectric grain boundaries that act as potential barriers for electrons [27,28,29,30]. In the FIT model, tunneling occurs between large metallic grains across insulating barriers with width w and area A. FIT is specified by two temperatures: temperature T₁ and a normalized temperature, T₀. T₁ characterizes the electrostatic energy of the parabolic potential barrier as follows:

$$k{T}_1={\displaystyle \frac{A\cdot w\cdot {\epsilon }_o{E}_0^2}{2}}.$$

(6)

Here, k is the Boltzmann constant, and ε₀ is the vacuum permittivity.

The characteristic electric field strength E₀ is determined by the barrier height V₀ as follows:

$$E_0={\displaystyle \frac{4V_0}{e\cdot w}},$$

(7)

where e is the electron charge. The normalized temperature T₀ relates T₁ to the tunneling constant as follows:

$$T_0=T_1{\left({\displaystyle \frac{\pi \chi w}{2}}\right)}^{-1},$$

(8)

with the reciprocal localization length of the wave function

$$\chi ={\displaystyle \frac{\sqrt{{2m_e}^{*}{V}_0}}{ħ}},$$

(9)

where m*_e is the effective electron mass, which in our case is m*_e = 2.5 m_e with m_e being the electron mass [31]. At elevated temperatures (above the resistivity minimum), the temperature dependencies of the specific electrical resistance ρ of SFMO (cf. samples SFMO-73, SFMO-85 and SFMO 92) are characterized by a positive slope, which suggests metallic behavior. Below 500 K, single-crystalline half-metallic SFMO exhibits a resistivity, ρ(T), that increases with temperature:

$$\rho (T)={\rho }_0+R_2{T}^2,$$

(10)

where ρ₀ = 1.8 × 10⁻⁶ Ωm is the residual metallic resistivity and R₂ = 2.5 × 10⁻¹⁰ ΩmK⁻² [32,33]. Similar values, ρ₀ = 3.28 × 10⁻⁶ Ωm and R₂ = 3.116 × 10⁻¹⁰ ΩmK⁻², were given in [34].

The metallic resistivity of SFMO originates from the Fe-O-Mo double exchange, where electrons hop between Fe and Mo via oxygen [35]. However, the presence of Fe-O-Fe bonds causes antiferromagnetic superexchange interactions because Fe spins are oppositely aligned. This hinders itinerant electron hopping because conduction “dead ends” are formed in the Fe–O–Mo conduction network [36]. Experimentally, the conductivity (the inverse of resistivity) σ varies from 16 S·cm⁻¹ for Sr₂Fe_1.4Mo_0.6O_6−δ to 75 S·cm⁻¹ for Sr₂Fe_1.6Mo_0.4O_6−δ [37]. In addition, the conductivity of Sr₂Fe_1.5Mo_0.5O_6−δ is much lower than that of SFMO under similar conditions [38]. At first glance, this creates a contradiction: the material exhibits high conductivity despite the presence of bonds that should suppress it. This contradiction is resolved within the framework of the short-range order and antiphase boundary models. Experimental XAFS data show that even with significant long-range disorder, a high degree of local ordering of the Fe–O–Mo bonds is retained [12]. Consequently, the conductivity is determined precisely by the short-range order, and the Fe–O–Fe bonds are concentrated at the domain boundaries and play the role of local defects. In contrast, the energetic advantage of the Fe–O–Mo double exchange over the Fe–O–Fe superexchange ensures the dominance of the conducting channel, which allows the material to retain its metallic character and high spin polarization.

Merging Equations (5) and (10) enables a consistent description of the SFMO resistivity across the experimental temperature range of 5–300 K. Consequently, the total resistivity is given by [25]

$$\rho (T)={\rho }_0+R_m{T}^m+{\rho }_{FIT}\exp \left({\displaystyle \frac{T_1}{T_0+T}}\right).$$

(11)

In contrast to single-crystalline materials, ceramics exhibit a microstructure consisting of interfaces, pores, grains, and grain boundaries. Consequently, the impact of antisite defects on the physical properties depends on the localization of the defect. While there are a number of studies on single crystals, such as SnSe [39], transparent semiconducting oxides [40], and alkaline-earth-metal hexaborides [41], as well as a first-principles study of double perovskite Pb₂FeOsO₆ [42], research on ceramics is still scarce.

In this study, we examined the influence of ASD on the resistivity of SFMO ceramics fabricated using the solid-state reaction method. We demonstrate that in granular SFMO ceramics, ASD affects the effective barrier height of intergrain tunneling and the effective values of barrier width and barrier area.

2. Materials and Methods

SFMO was synthesized using a conventional solid-state reaction method involving three successive annealing stages. The initial mixture was prepared from SrCO₃, Fe₂O₃, and MoO₃ powders taken in a stoichiometric ratio of 2:0.5:1. The precursors were thoroughly mixed and ground in a ball mill PM 100 (Retsch GmbH, Haan, Germany) to ensure uniform mixing. The first annealing was performed in air at T = 1173 K for 20 h. The resulting powder was subsequently reground and additionally mixed in ethyl alcohol for 2 h to improve homogeneity. The second annealing step was performed in an argon atmosphere at T = 1373 K for 20 h. The obtained SFMO powder was then compacted into cylindrical pellets (10 mm in diameter, 5–6 mm in thickness) under a uniaxial pressure of 250 MPa at room temperature. Finally, the pellets were sintered in a reducing atmosphere (10% H₂/Ar) at T = 1373 K for different durations (10, 14, 19, 27 and 40 h). In the final step, SFMO with varying ASD values was obtained, given that ASD depends on both annealing time and temperature under reducing conditions [43,44].

The phase composition of solid-state synthesis products as well as of the SFMO samples was determined using XRD phase analysis on a DRON-3 apparatus (PA “Lvivselmash”, Lviv, USSR) in CuK_α radiation. Rietveld refinement was performed using PowderCell v.2.3 software [45] with the ICSD-PDF2 database (Release 2000).

The magnetic and electrical transport properties of the samples were studied in the temperature range of 5 K to 300 K using a constant magnetic field in a universal Cryogen-Free High-Magnetic Field Measurement System Vibrating Sample Magnetometer CFM-14T-H3-IVTI-25 (Cryogenic Ltd., London, UK).

3. Results and Discussion

The XRD patterns of the samples considered in this work are depicted in Figure 2. All SFMO samples are single-phase. The presence of (101) and (103) reflexes indicates the appearance of superstructural ordering of Fe and Mo ions in the SFMO unit cell. For readers’ convenience, sample names in this work include the value of the order parameter P.

Another type of defect in SFMO that affects the electron transport properties, magnetization, and magnetoresistance is the oxygen vacancy [46]. The oxygen vacancy content in SFMO is characterized by the oxygen nonstoichiometry parameter δ. This parameter can be evaluated using thermogravimetric analysis, iodometric titration, and neutron diffraction. In addition, δ can be estimated using XRD, although this approach is associated with large uncertainty. Following the data reported in [47], high-temperature annealing of SFMO in a reducing atmosphere stabilizes the parameter δ reaching a constant value after approximately 10 h of thermal treatment. In the present study, all SFMO samples were annealed in a reducing atmosphere for 10–40 h, ensuring that δ attained the same equilibrium value for all samples. This implies that oxygen nonstoichiometry has an identical impact on the electron transport properties of the investigated samples. Thus, in the following analysis, the contribution of δ is not considered further, and attention is focused on the role of ASD in determining the transport behavior. On the other hand, the sintering temperature in this work is well below the temperature required for sufficient grain growth in perovskite SrTiO₃ ceramics [48]. Consequently, SEM images did not give evidence for changes in ceramic microstructure during sintering.

Figure 3 illustrates the magnetic field dependence of the magnetization of the samples on the order parameter P at T = 5 K.

Table 1. Site occupation factors g defined from the Rietveld refinement, the order parameter P calculated according to Equation (2), and ASD resulting from Equation (1) and the order parameter P_M derived from the ASD of Equation (3) for b = 8. The saturated magnetization values of the SFMO samples were determined using data from Figure 2.

Sample	gFe	gMo	P, %	ASD	Ms, MB/f.u.	PM, %
SFMO-57	0.785	0.785	57	0.215	2.33	58.25
SFMO-73	0.865	0.865	73	0.135	2.89	72.25
SFMO-85	0.925	0.925	85	0.075	3.36	84
SFMO-92	0.960	0.960	92	0.040	3.67	91.75

compiles the order parameter P, antisite disorder, and the corresponding site occupation factors of the B′ and B″ sublattices, g_Fe and g_Mo, defined from the Rietveld refinement, the order parameter P calculated according to Equation (2), and ASD resulting from Equation (1) and the order parameter P_M derived from the ASD of Equation (3). With regard to differences in b values, the uncertainty of the P_M value amounts to ±15% for M_s = 2.33 µB, decreasing to ±2% for M_s = 3.67 µ_B/f.u..

To determine the charge carrier scattering mechanisms in the high-temperature range, graphs of the ρ(T) ~ ρ₀ + R_mT^m type were drawn for the temperature dependence of the SFMO resistivity. The best power-law approximation was a value of m = 2, obtained in a limited temperature range, as illustrated in Figure 4.

The fitted value m ≈ 2 probably corresponds to electron–electron scattering (characteristic of transition metal oxide systems, which include SFMO) or alternatively to electron scattering by phonons [49] or by magnons [50,51,52]. Electron–electron scattering due to Coulomb interactions between free charge carriers makes a very weak contribution to the resistivity of SFMO because of the low effective charge carrier density at the Fermi level, equal to 1.1 electron/f.u. at T = 20 K [34] and 1.3 electron/f.u. at T = 300 K [53]. In addition, the contribution of electron scattering leads to a positive magnetoresistance [54], which is observed, for instance, in metallic alloys, whereas SFMO has a negative MR. Another mechanism that explains the quadratic dependence ρ ~ T² is the scattering of conduction electrons by phonons, that is, quanta of vibrational motion of atoms or ions in the crystal lattice [55]. Under the influence of an external electric field, electrons move through the crystal lattice and collide with phonons, exchanging energy and momentum, which alters their trajectories. As the temperature increases, the amplitude of the thermal vibrations of atoms increases, thereby increasing the frequency of such collisions. Consequently, the electron–phonon interaction limits electron mobility, thereby increasing the electrical resistance of the material [56]. At low temperatures, this scattering mechanism is weakened, allowing electrons to move with less energy loss; therefore, as the temperature decreases, the resistivity decreases, as shown in the Figure 4.

The observed decrease in the R_m factor (cf.

Table 2. Fitting resistivity model parameters for samples in this work.

Sample	ρ0, Ω·m	Rm, Ω·m·K−2	ρFIT, mΩ	T1, K	T0, K
SFMO-57	-	-	5.46 × 10−4	145.34	135.28
SFMO-73	3.46 × 10−3	1.31 × 10−9	1.26 × 10−4	120.19	130.05
SFMO-85	3.39 × 10−4	9.65 × 10−10	7.50 × 10−4	95.08	120.01
SFMO-92	3.32 × 10−4	8.72 × 10−10	3.26 × 10−4	70.11	104.97
SFMO-98	2.56 × 10−4	7.93 × 10−10	-	-	-

) with an increasing degree of superstructural ordering of Fe/Mo indicates a weakening of the contribution of electron-magnon scattering to the high-temperature part of the SFMO [57,58]. In double perovskite SFMO, ideal superstructural ordering (Fe/Mo) creates a periodic lattice potential. This ensures high carrier mobility in subbands with a specific spin orientation (half-metallic state). Spin waves (magnons) have a clearly defined spectrum. Electron-magnon scattering in such a system is relatively weak, since it is difficult for conduction electrons with a given spin polarization to interact with collective excitations in a strictly ordered spin system. Antisite defects locally disrupt the periodicity of the crystalline and magnetic order. This leads to a local distortion of the electric potential because the Fe³⁺ and Mo^5+/6+ ions have different charges and electron configurations. This creates a region of non-uniform scattering potential for conduction electrons. Fe³⁺–O–Fe³⁺ or Mo–O–Mo defects also appear, which locally alter the type and strength of magnetic interaction, creating magnetic inhomogeneities (violation of spin collinearity, weakening of exchange interaction). Electron-magnon scattering is enhanced in an inhomogeneous medium. Antisite defects act as centers at which conduction electrons interact more effectively with the spin system: an electron passing a magnetic defect can induce a localized spin excitation or change its spin orientation. Low-energy magnons in a disordered system have a higher density of states and can more easily exchange energy with electrons. Thus, the more antisite defects (the lower the degree of ordering P), the stronger the overall scattering of conduction electrons. At a high degree of ordering, the periodic potential maintains regularity, which suppresses electron-magnon scattering and minimizes its contribution to electrical resistance. On the contrary, at low ordering, many antisite defects arise, causing local disturbances in the electronic and magnetic order and forming inhomogeneities. As a result, electron-magnon scattering becomes an effective channel for electron scattering, and its contribution to resistance increases significantly. In this way, antisite defects serve as centers that enhance the interaction between electron and spin degrees of freedom, which is experimentally manifested as an increase in the coefficient R_m in the temperature dependence of the resistivity.

Figure 5 illustrates the resistivity data of the already considered polycrystalline SFMO ceramic samples fabricated by the solid-state reaction technique. With regard to the large number of fitting parameters, first the high temperature part of ρ(T) was fitted to Equation (10). Then, using these results as starting parameters, the parameters of the FIT model were obtained by a fit over the entire temperature range using Equation (11), shown in Figure 5 by solid lines.

Table 2 gathers the fitting parameters for the model described by Equations (5) and (11) and the special case ρ(T) = ρ₀ + R_mT² for sample SFMO-98. Figure 4 and Table 2 reveal that the minimum resistivity sufficiently depends on the order parameter P. Note that in the considered temperature range, the moderately disordered sample SFMO-57 is well described solely by the FIT model, Equation (5).

The residual resistivity values obtained for polycrystalline SFMO ceramics exceed those of single-crystal SFMO by approximately two orders of magnitude [52], indicating that grain boundary scattering is the primary mechanism controlling charge carrier transport in the polycrystalline system. In this work, we consider intergrain charge carrier transport by fluctuation-induced tunneling. Following Equation (5), the thermal activation energy of charge transport in the high-temperature region, T >> T₀, is given by

$$E_a=k{T}_1.$$

(12)

The value of E_a can be considered an effective value of the tunneling barrier height eV₀. It allows us to calculate parameters of the FIT model as follows:

$$\begin{array}{c}w={\displaystyle \frac{ħ}{\pi T_0}}\sqrt{{\displaystyle \frac{2T_1}{m_e^{*}k}}},\\ A={\displaystyle \frac{k{T}_1{e}^{2}w}{8{\epsilon }_0{V}_0^2}}={\displaystyle \frac{e^{2}w}{8{\epsilon }_0{E}_a}}.\end{array}$$

(13)

Figure 6 shows the dependence of the effective tunneling barrier height E_a, Equation (12), the barrier width w and barrier area A (calculated by means of Equation (13)) on the ASD values.

In granular ceramics, grain boundaries are the main source of conductivity loss compared to single crystals, i.e., they are the resistive bottleneck. Typically, a disordered grain boundary in ceramics increases resistivity because it introduces structural irregularities, defect states, and potential barriers that trap electrons and, thereby, hinder charge carrier transport. Structural irregularities, localized trap states and trapped charges at the boundary increase the potential barrier that must be overcome by charge carriers. Disorder at grain boundaries increases this potential barrier for intergrain tunneling even more. Thus, we expect an increase in E_A with ASD, which corresponds in our case to an increase in T₁. The increase in E_A entails a decrease in the effective area of the nanosized barriers (cf. Equation (13)).

4. Conclusions

In this work, we considered the influence of antisite disorder, e.g., Fe ions on Mo sites, Fe_Mo, and vice versa, Mo_Fe, on the resistivity of strontium ferromolybdate ceramics fabricated by the solid-state reaction method.

We demonstrate that antisite disorder affects the parameters of the nanosized intergrain tunneling barriers, leading to fluctuation-induced tunneling at low temperatures. Both the effective barrier height of intergrain tunneling and the effective barrier width increase with increasing disorder, while the effective barrier area decreases in accordance with its dependence on effective barrier height.